On the Well-posedness of 2-D Incompressible Navier-Stokes Equations with Variable Viscosity in Critical Spaces

الأصل · EN

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier-Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for p∈(1,4) and a∈Bₚ,₁2p(R²) that the solution mapping Hₐ:F∇Π to the 2-D elliptic equation div((1+a)∇Π)=div F is bounded on Bₚ,₁2p-1(R²). More precisely, we prove that ∇ΠBₚ,₁2p-1≤ C(1+aBₚ,₁2p)²FBₚ,₁2p-1. The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15]-[17]. When the viscosity coefficient μ(ρ) is a positive constant, we prove that (1.2) is globally well-posed.